Abstract
We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal–dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring, and matrix factorization are provided.
Highlights
The problem at hand is the following structured convex optimization problem min f (x) + g(K x), (1)x ∈H for real Hilbert spaces H and G, f : H → R := R ∪ {±∞} a proper, convex and lower semicontinuous function, g : G → R a, possibly nonsmooth, convex and Lipschitz continuous function, and K : H → G a linear continuous operator.Research partially supported by FWF (Austrian Science Fund) project I 2419-N32
Research supported by the doctoral programme Vienna Graduate School on Computational Optimization (VGSCO), FWF (Austrian Science Fund), Project W 1260
The approach can be described as follows: we “smooth” g, i.e. we replace it by its Moreau envelope, and solve the resulting optimization problem by an accelerated proximal-gradient algorithm
Summary
The problem at hand is the following structured convex optimization problem min f (x) + g(K x),. X ∈H for real Hilbert spaces H and G, f : H → R := R ∪ {±∞} a proper, convex and lower semicontinuous function, g : G → R a, possibly nonsmooth, convex and Lipschitz continuous function, and K : H → G a linear continuous operator. Research partially supported by FWF (Austrian Science Fund) project I 2419-N32. Research supported by the doctoral programme Vienna Graduate School on Computational Optimization (VGSCO), FWF (Austrian Science Fund), Project W 1260
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
